CN115900511B - Nonlinear separable least square-based magnetic dipole target positioning method - Google Patents
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Abstract
The invention relates to the technical field of magnetic target positioning, and discloses a magnetic dipole target positioning method based on nonlinear separable least square, which is characterized in that magnetic anomaly data generated by a magnetic target is obtained by utilizing a magnetic sensor array, then an optimization problem of magnetic target positioning is converted into a nonlinear separable least square problem, namely unknown parameters in magnetic target positioning are divided into linear and nonlinear two parts, the optimization problem is converted into a nonlinear optimization problem only comprising target position parameters by adopting a variable projection method, the number of the unknown parameters is reduced from six to three, and then the nonlinear optimization problem is solved by a particle swarm optimization algorithm, so that the position parameters of the target are obtained, and the target positioning is realized.
Description
Technical Field
The invention relates to the technical field of magnetic target positioning, in particular to a magnetic dipole target positioning method based on nonlinear separable least square.
Background
The magnetic target can generate a magnetic anomaly signal in the geomagnetic field, the magnetic anomaly signal can be obtained through magnetic sensor measurement, the strength of the signal intensity depends on the space relative position between the target and the magnetic sensor and the size of the magnetic moment of the target, so that the position of the target can be estimated from the magnetic anomaly signal through an inversion algorithm, the magnetic target positioning and tracking can be realized, and the magnetic target positioning and tracking method has important application value in the military and civil fields such as unexplosive object detection, vehicle positioning, capsule endoscope positioning and the like. When the distance between the magnetic target and the magnetic sensor is greater than 2.5 times of the target size, the magnetic anomaly signal generated by the target can be generally described by a magnetic dipole target positioning model, and the magnetic target positioning problem can be converted into a magnetic dipole target positioning model parameter estimation problem. At present, the problem is mainly solved by a parameter optimizing method, and mainly comprises two types of heuristic optimization algorithms and classical numerical optimization algorithms. In the prior art, a particle swarm optimization algorithm is combined with a Levenberg-Marquart (LM) algorithm and is used for positioning a magnetic target, and the method can solve the problem of initial value sensitivity of the traditional LM algorithm in solving nonlinear optimization. Although the above-mentioned optimization methods can realize magnetic target positioning and magnetic moment parameter estimation, these methods are all performed in six-dimensional space containing target position and magnetic moment parameter, so that the search efficiency is low, and local extremum is easy to be trapped, and the practicability is reduced.
The invention relates to a magnetic dipole target positioning method based on nonlinear separable least square, which utilizes a magnetic sensor array to acquire magnetic anomaly data generated by a magnetic target, then converts an optimization problem of magnetic target positioning into a nonlinear separable least square problem, namely, divides an unknown parameter in magnetic target positioning into linear and nonlinear two parts, converts the optimization problem into a nonlinear optimization problem only comprising target position parameters by adopting a variable projection method, reduces the number of the unknown parameters from six to three, and then solves the problem by a particle swarm optimization algorithm to obtain the position parameters of the target, thereby realizing target positioning.
Disclosure of Invention
In order to overcome the defects in the prior art, the invention provides a magnetic dipole target positioning method based on nonlinear separable least square, which converts a magnetic target positioning problem into a separable nonlinear least square problem by utilizing the separable characteristics of a target position parameter and a magnetic moment parameter, reduces unknown parameters from six to three by utilizing variable projection, and improves optimizing efficiency and positioning and parameter estimation precision when solving the nonlinear least square problem by utilizing a particle swarm optimization algorithm.
The technical aim of the invention is realized by the following technical scheme: a magnetic dipole target positioning method based on nonlinear separable least squares comprises the following steps:
S1, the magnetic target is equivalent to a magnetic dipole target positioning model, magnetic abnormal data generated by the magnetic target in motion are received by utilizing a magnetic sensor, and the specific method is as follows:
a. establishing a Cartesian coordinate system, setting a magnetic sensor coordinate as B (x, y, z), setting a magnetic target coordinate as A (x 0,y0,z0), and setting a magnetic moment vector as m (m x,my,mz), wherein m x,my,mz respectively represents projection of the magnetic moment vector of the target on each coordinate axis;
When the distance between the magnetic target and the magnetic sensor is more than 2.5 times of the size of the magnetic sensor, a magnetic dipole target positioning model is established, and the expression is as follows:
b. expanding the above formula (1) into a matrix expression as follows:
Wherein B is the magnetic induction generated by the magnetic target received by the magnetic sensor at the point B, B x,By,Bz is the projection of the magnetic induction B on each coordinate axis, mu 0 is the magnetic permeability in vacuum, the size of which is 4pi× -7H/m,r＝(x-x0,y-y0,z-z0) is the relative direction vector between the target and the sensor, Representing the distance between the target and the sensor;
c. The above formula (2) may be further expressed as:
B＝FM(3)
wherein,
As can be obtained from the above formula (3), the magnetic anomaly generated by the magnetic target is composed of two parts, F and M, where F is a matrix related to the three-dimensional relative position of the magnetic target and the magnetic sensor, and M is a vector related to the magnetic moment of the magnetic target;
s2, forming a magnetic sensor array through a plurality of magnetic sensors, and constructing a separable nonlinear least square model by utilizing magnetic anomaly data received by the magnetic sensor array, wherein the specific method comprises the following steps of:
a. consider the condition of multiple magnetic sensor observations:
Ball＝Fall M(4)
Wherein ,Ball＝[B1;B2;...;BN],Fall＝[F1;F2;...;FN],N is the number of magnetic sensors, B i (i=1, 2, …, N) represents the magnetic induction vector obtained by the ith magnetic sensor, and F i (i=1, 2, …, N) represents a matrix formed by the relative positions of the magnetic target and the ith magnetic sensor;
b. Converting the magnetic target positioning and magnetic moment parameter estimation problem into a nonlinear least squares problem as described by the following equation:
Wherein p= (x, y, z) represents a vector to be estimated constituted by the magnetic target position;
c. In the case where the number of magnetic sensors is known, the problem of estimating the target magnetic moment vector M is converted into a linear least squares problem, expressed as follows:
M＝(Fall TFall)-1Fall TBall(6)
d. Substituting the above formula (6) into formula (5) to obtain:
s3, solving the formula (7) by adopting a particle swarm optimization algorithm to obtain a target position;
S4, calculating F all according to the target position obtained in the step S3, and calculating a magnetic moment vector parameter M according to the formula (6).
Further, the magnetic sensor is a three-axis fluxgate sensor.
Further, the particle swarm optimization algorithm described in step S3 specifically includes the following steps:
Step 1: particle swarm initialization: setting parameters such as particle number, learning factors, maximum and minimum inertia weights, iteration times and the like; and randomly generating L particles x i and corresponding particle speeds v i in the set search space;
Step 2: calculating the fitness function value of each particle;
Step 3: comparing the fitness of the particles with the previous fitness of the individual extremum, and selecting a relatively better position to update the individual extremum;
step 4: comparing the fitness of the particles with the previous global optimal fitness, and selecting a relatively better position to update a global extremum;
step 5: judging whether the maximum iteration step number is reached, and if the maximum iteration step number is reached, outputting an optimizing result, namely a target position; otherwise, jump to step 2.
In summary, the invention has the following beneficial effects: according to the invention, the magnetic target positioning problem is converted into the separable nonlinear least square problem by utilizing the separable characteristics of the target position parameter and the magnetic moment parameter, the unknown parameters are reduced from six to three by utilizing variable projection, and the optimizing efficiency can be improved when the nonlinear least square problem is solved by utilizing a particle swarm optimization algorithm, and the positioning and parameter estimation accuracy is improved.
Drawings
FIG. 1 is a flow chart of a magnetic dipole table positioning method based on separable nonlinear least squares model and particle swarm optimization algorithm in an embodiment of the invention;
FIG. 2 is a schematic diagram of the geometric relationship between a target and a magnetic sensor in an embodiment of the invention;
FIG. 3 is a schematic diagram of a top view of the relative positions of a target and a magnetic sensor array in an embodiment of the invention.
Detailed Description
The invention is described in further detail below with reference to fig. 1-3.
Examples: a nonlinear separable least squares-based magnetic dipole target positioning method, as shown in fig. 1, comprises the following steps:
S1, the magnetic target is equivalent to a magnetic dipole target positioning model, magnetic abnormal data generated by the magnetic target in motion are received by utilizing a magnetic sensor, and the specific method is as follows:
a. Establishing a Cartesian coordinate system, as shown in FIG. 2, which is a schematic diagram of the geometric relationship between a target and a magnetic sensor, wherein the magnetic sensor coordinate is B (x, y, z), the magnetic target coordinate is A (x 0,y0,z0), and the magnetic moment vector is m (m x,my,mz), wherein m x,my,mz respectively represents the projection of the magnetic moment vector of the target on each coordinate axis;
When the distance between the magnetic target and the magnetic sensor is more than 2.5 times of the size of the magnetic sensor, a magnetic dipole target positioning model is established, and the expression is as follows:
b. expanding the above formula (1) into a matrix expression as follows:
Wherein B is the magnetic induction generated by the magnetic target received by the magnetic sensor at the point B, B x,By,Bz is the projection of the magnetic induction B on each coordinate axis, mu 0 is the magnetic permeability in vacuum, the size of which is 4pi× -7H/m,r＝(x-x0,y-y0,z-z0) is the relative direction vector between the target and the sensor, Representing the distance between the target and the sensor;
c. The above formula (2) may be further expressed as:
B＝FM(3)
wherein,
As can be obtained from the above formula (3), the magnetic anomaly generated by the magnetic target is composed of two parts, F and M, where F is a matrix related to the three-dimensional relative position of the magnetic target and the magnetic sensor, and M is a vector related to the magnetic moment of the magnetic target;
s2, forming a magnetic sensor array through a plurality of magnetic sensors, and constructing a separable nonlinear least square model by utilizing magnetic anomaly data received by the magnetic sensor array, wherein the specific method comprises the following steps of:
a. consider the condition of multiple magnetic sensor observations:
Ball＝FallM (4)
Wherein ,Ball＝[B1;B2;...;BN],Fall＝[F1;F2;...;FN],N is the number of magnetic sensors, B i (i=1, 2, …, N) represents the magnetic induction vector obtained by the ith magnetic sensor, and F i (i=1, 2, …, N) represents a matrix formed by the relative positions of the magnetic target and the ith magnetic sensor;
b. Converting the magnetic target positioning and magnetic moment parameter estimation problem into a nonlinear least squares problem as described by the following equation:
Wherein p= (x, y, z) represents a vector to be estimated constituted by the magnetic target position;
c. In the case where the number of magnetic sensors is known, the problem of estimating the target magnetic moment vector M is converted into a linear least squares problem, expressed as follows:
M＝(Fall TFall)-1Fall TBall (6)
d. Substituting the above formula (6) into formula (5) to obtain:
s3, solving the formula (7) by adopting a particle swarm optimization algorithm to obtain a target position, wherein the method specifically comprises the following steps:
Step 1: particle swarm initialization: setting parameters such as particle number, learning factors, maximum and minimum inertia weights, iteration times and the like; and randomly generating L particles x i and corresponding particle speeds v i in the set search space;
Step 2: calculating the fitness function value of each particle;
Step 3: comparing the fitness of the particles with the previous fitness of the individual extremum, and selecting a relatively better position to update the individual extremum;
step 4: comparing the fitness of the particles with the previous global optimal fitness, and selecting a relatively better position to update a global extremum;
step 5: judging whether the maximum iteration step number is reached, and if the maximum iteration step number is reached, outputting an optimizing result, namely a target position; otherwise, jump to step 2.
The particle swarm optimization algorithm is a heuristic intelligent swarm optimization algorithm, and the mathematical description is as follows: the method is characterized in that the group is composed of L particles, the particles fly at a certain speed in a D-dimensional search space, and when each particle searches, the self-searched historical optimal position (Prest) and the historical optimal positions (Gbest) of other particles in the group are considered, and the particle positions are updated on the basis until the fitness function is minimum. Let the position and velocity of the ith particle of the particle swarm in the D-dimensional search space be: x i＝(xi1,xi2,…,xiD) and v i＝(vi1,vi2,…,viD), the speed and position of each particle are updated by means of the individual and global optimum. The individual optimal value P i＝[pi1,pi2,…,piD]T, the adaptation value of the adaptation function is P besti, the population extremum represents the optimal position P g＝[pg1,pg2,…,pgD]T searched by the particle swarm, and the corresponding adaptation value is g besti. The update formulas of the d-th dimensional speed and the position of the particle i are respectively as follows:
wherein k is the iteration number, ω is the inertial weight, c 1 and c 2 are the learning factors, r 1 and r 2 are the random numbers distributed over [0,1], For the d-th dimension component of the velocity vector of the kth iterative particle i,/>Is the d-th dimension component of the position vector of the particle i for the kth iteration.
S4, calculating F all according to the target position obtained in the step S3, and calculating a magnetic moment vector parameter M according to the formula (6).
The following is the content of the simulation analysis of the invention:
Assuming that the magnetic moment of the target is m (m x,my,mz)＝(500,300,200)A·m2, positions are (10 m, -5m,2 m) & lt three-axis fluxgate sensors are (0 m,0 m), (3 m,0 m) and (6 m,0 m) respectively, and the directions of the magnetic sensors are the same as the directions of the target-sensor coordinate system, the top view of the spatial position relationship between the target and the magnetic sensor array is shown in fig. 3. Fig. 3 is a schematic diagram of the top view of the relative positions of the target and the magnetic sensor array, the magnetic sensor is inevitably affected by measurement noise in the measurement process, the average value of the measurement noise in each axial direction is assumed to be 0, and the standard deviation is 0.1nT. The separable nonlinear least square model-based particle swarm optimization algorithm and the conventional particle swarm optimization algorithm set forth herein are adopted to process the magnetic sensor array measurement data, wherein the particle swarm optimization algorithm is set to 100, the maximum iteration times are set to be 50, the learning factors are set to be 2, the maximum and the inertia weights are respectively 0.9 and 0.2, and the two positioning methods are statistically shown in table 1 below:
table 1 statistics of target parameter estimation results for different methods
When the magnetic target positioning and parameter estimation are carried out by using the conventional particle swarm optimization algorithm from the results shown in the table 1, the target position estimation is closer to the real position, but the estimation result of the magnetic moment parameter has larger deviation from the true value, because the conventional particle swarm optimization algorithm carries out optimization on the target position and the magnetic moment six-dimensional parameter space, the objective function has more local extremum points, and the particle swarm optimization algorithm is difficult to simultaneously and accurately estimate six unknown parameters; the target parameter result obtained by the method provided by the invention is very similar to the true value, which shows that the method can more accurately realize target positioning and magnetic moment parameter estimation, because the method reduces the dimension of an unknown parameter space of an optimization problem by a variable projection method, the number of the unknown parameters is reduced from six to three, the possibility that the algorithm falls into a local extremum is reduced, the robustness of the positioning algorithm is improved, and the precision of parameter estimation is also improved. Meanwhile, from the viewpoint of operation time, the average operation time of a conventional particle swarm optimization algorithm is 0.13s, and the average operation time of the method disclosed by the invention is 0.11s, and the calculation efficiency of the algorithm is improved by reducing the search space dimension of unknown parameters.
The effects of different sensor measurement noise levels and different sensor spacing on the target positioning effect are discussed below. Assuming that the measured noise levels of the sensors are 0.01nT and 1nT, respectively, the target location results using the methods herein are shown in table 2 below:
| Parameters (parameters) | x | y | z | mx | my | mz |
| True value | 10 | -5 | 2 | 500 | 300 | 200 |
| 1nT | 9.998 | -4.990 | 1.987 | 498.55 | 298.86 | 198.09 |
| 10nT | 9.762 | -4.766 | 1.691 | 508.15 | 338.04 | 163.37 |
TABLE 2 statistics of estimation results of target parameters under different noise levels
As can be seen from the results of table 2 above, when the measurement noise of the magnetic sensor is small (1 nT or 0.1 nT), the target parameter estimation result is very close to the true value; if the measurement noise is large (10 nT), the estimation error of the target parameter, especially the magnetic moment parameter of the target, is large, although the method can still converge. Therefore, in order to improve the target positioning effect, it is necessary to reduce the measurement noise of the magnetic sensor as much as possible.
Next, the measured noise level of the sensor was kept at 0.1nT, the spacing between the three sensors was varied, and the effectiveness of the methods presented herein under different sensor spacing conditions was tested. Assuming that the position of the first sensor remains unchanged, the pitch of the sensors becomes 1m (abbreviated as pitch one) and 5m (abbreviated as pitch two), respectively, and the target parameter estimation results obtained using the method presented herein are shown in table 3 below.
| Parameters (parameters) | x | y | z | mx | my | mz |
| True value | 10 | -5 | 2 | 500 | 300 | 200 |
| Spacing 1m | 10.045 | -4.998 | 1.871 | 506.05 | 303.68 | 188.08 |
| Spacing 5m | 9.998 | -5.001 | 2.002 | 500.39 | 300.26 | 200.01 |
TABLE 3 estimation results of target parameters under different sensor spacing conditions
As can be seen from the results of table 3 above, the target positioning effect under the condition of the first sensor pitch is slightly worse than that of the second sensor pitch, that is, increasing the sensor pitch is beneficial to improving the accuracy of target positioning and parameter estimation.
In summary, according to the magnetic target positioning and parameter estimation problems, the magnetic target positioning problem is converted into the separable nonlinear least square problem by utilizing the separable characteristics of the target position parameters and the magnetic moment parameters, unknown parameters are reduced from six to three by utilizing variable projection, and the optimizing efficiency can be improved when the nonlinear least square problem is solved by utilizing a particle swarm optimization algorithm, and the positioning and parameter estimation accuracy is improved.
The present embodiment is only for explanation of the present invention and is not to be construed as limiting the present invention, and modifications to the present embodiment, which may not creatively contribute to the present invention as required by those skilled in the art after reading the present specification, are all protected by patent laws within the scope of claims of the present invention.
Claims (3)
1. A method for locating a magnetic dipole target based on nonlinear separable least squares, comprising the steps of:
S1, the magnetic target is equivalent to a magnetic dipole target positioning model, magnetic abnormal data generated by the magnetic target in motion are received by utilizing a magnetic sensor, and the specific method is as follows:
a. establishing a Cartesian coordinate system, setting a magnetic sensor coordinate as B (x, y, z), setting a magnetic target coordinate as A (x 0,y0,z0), and setting a magnetic moment vector as m (m x,my,mz), wherein m x,my,mz respectively represents projection of the magnetic moment vector of the target on each coordinate axis;
When the distance between the magnetic target and the magnetic sensor is more than 2.5 times of the size of the magnetic sensor, a magnetic dipole target positioning model is established, and the expression is as follows:
b. expanding the above formula (1) into a matrix expression as follows:
Wherein B is the magnetic induction generated by the magnetic target received by the magnetic sensor at the point B, B x,By,Bz is the projection of the magnetic induction B on each coordinate axis, mu 0 is the magnetic permeability in vacuum, the size of which is 4pi× -7H/m,r＝(x-x0,y-y0,z-z0) is the relative direction vector between the target and the sensor, Representing the distance between the target and the sensor;
c. The above formula (2) may be further expressed as:
B＝FM (3)
wherein,
As can be obtained from the above formula (3), the magnetic anomaly generated by the magnetic target is composed of two parts, F and M, where F is a matrix related to the three-dimensional relative position of the magnetic target and the magnetic sensor, and M is a vector related to the magnetic moment of the magnetic target;
s2, forming a magnetic sensor array through a plurality of magnetic sensors, and constructing a separable nonlinear least square model by utilizing magnetic anomaly data received by the magnetic sensor array, wherein the specific method comprises the following steps of:
a. consider the condition of multiple magnetic sensor observations:
Ball＝FallM (4)
Wherein ,Ball＝[B1;B2;...;BN],Fall＝[F1;F2;...;FN],N is the number of magnetic sensors, B i (i=1, 2, …, N) represents the magnetic induction vector obtained by the ith magnetic sensor, and F i (i=1, 2, …, N) represents a matrix formed by the relative positions of the magnetic target and the ith magnetic sensor;
b. Converting the magnetic target positioning and magnetic moment parameter estimation problem into a nonlinear least squares problem as described by the following equation:
Wherein p= (x, y, z) represents a vector to be estimated constituted by the magnetic target position;
c. In the case where the number of magnetic sensors is known, the problem of estimating the target magnetic moment vector M is converted into a linear least squares problem, expressed as follows:
M＝(Fall TFall)-1Fall TBall (6)
d. Substituting the above formula (6) into formula (5) to obtain:
s3, solving the formula (7) by adopting a particle swarm optimization algorithm to obtain a target position;
S4, calculating F all according to the target position obtained in the step S3, and calculating a magnetic moment vector parameter M according to the formula (6).
2. The method of claim 1, wherein the magnetic sensor is a three-axis fluxgate sensor.
3. The method for positioning a magnetic dipole target based on nonlinear separable least squares according to claim 1, wherein the particle swarm optimization algorithm of step S3 specifically comprises the following steps:
Step 1: particle swarm initialization: setting parameters such as particle number, learning factors, maximum and minimum inertia weights, iteration times and the like; and randomly generating L particles x i and corresponding particle speeds v i in the set search space;
Step 2: calculating the fitness function value of each particle;
Step 3: comparing the fitness of the particles with the previous fitness of the individual extremum, and selecting a relatively better position to update the individual extremum;
step 4: comparing the fitness of the particles with the previous global optimal fitness, and selecting a relatively better position to update a global extremum;
step 5: judging whether the maximum iteration step number is reached, and if the maximum iteration step number is reached, outputting an optimizing result, namely a target position; otherwise, jump to step 2.
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